Finite Group Databasebeta

← #10690324

Subgroups — #10690324

Order 768.

1050 conjugacy classes of subgroups.

The subgroup lattice has 720 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2325C211yesmaximal, characteristic, complemented
2C2×#5036C211yesmaximal, characteristic
3C2×#4537C211yesmaximal, characteristic, complemented
4#69521C211yesmaximal, characteristic, complemented
5#69426C211yesmaximal, characteristic, complemented
6#67624C211yesmaximal, characteristic, complemented
7#67613C211yesmaximal, characteristic, complemented
8#9041—31nomaximal, Sylow
9C23×D12C2211yescharacteristic
10C2×C3×#121C2211yescharacteristic
11C2×C3×C4×D4—21no—
12C22×#213—21no—
13C22×(C2×D3)⋊C4—21no—
14C22×(C2×D3)⋊C4—21no—
15C3×#388—21no—
16C3×#387—21no—
17C3×#386—21no—
18C3×#386—21no—
19C3×#385—21no—
20C3×#380C2211yescharacteristic
21C2×C3×#95C2211yescharacteristic
22C2×C3⋊#95C2211yescharacteristic
23C2×#830C2211yescharacteristic
24C2×#830C2211yescharacteristic
25#5036—22no—
26#5036C414yescomplemented
27C2×#809—21no—
28C2×#809—21no—
29C2×#809—21no—
30#4537—22no—
31C2×#447—31no—
32#2325D311yescharacteristic, complemented
33#1786—31no—
34C2×#327—31no—
35#1438—31no—
36#1294—31no—
37#1280—31no—
38C24×D3—21no—
39C22×C3×D4—21no—
40C22×C3×D4D411yescharacteristic
41C23×C3×C4—21no—
42C23×C3×C4D411yescharacteristic
43C23×C3×C4D411yescharacteristic
44C23×C3×C4C2311yescharacteristic
45C23×Dic3—21no—
46C22×D12—21no—
47C22×D12—21no—
48C22×D12—21no—
49C22×D12—21no—
50C22×D12—21no—
51C22×D12—42no—
52C22×D12—42no—
53C22×D12C2×C412yescomplemented
54C3×#121—21no—
55C3×#121—21no—
56C3×#121C2×C412yes—
57C3×#121D412yes—
58C3×C4×D4—42no—
59C3×C4×D4—42no—
60C2×C3×C4⋊C4—21no—
61C2×C3×C4⋊C4D411yescharacteristic
62C2×C3×C4⋊C4D411yescharacteristic
63C2×C3×C4⋊C4D411yescharacteristic
64C2×C3×C22⋊C4—21no—
65C2×C3×C22⋊C4—21no—
66C2×C3×C22⋊C4—21no—
67C2×C3×C22⋊C4—21no—
68C2×C3×C22⋊C4—21no—
69C2×C3×C42—21no—
70C2×#213—24no—
71C2×#213—42no—
72C2×#213—42no—
73C2×(C2×D3)⋊C4—42no—
74C2×(C2×D3)⋊C4—42no—
75C2×(C2×D3)⋊C4—42no—
76C2×(C2×D3)⋊C4—42no—
77C2×(C2×D3)⋊C4—42no—
78C2×(C2×D3)⋊C4—42no—
79C22×C3⋊C8—21no—
80C22×C3⋊C8—21no—
81C3×#95—21no—
82C3×#95—21no—
83C3×#95—21no—
84C3×#95—21no—
85C3×#95—21no—
86C3⋊#95—21no—
87C3⋊#95—21no—
88#830—21no—
89#830—21no—
90#830—21no—
91#830C2×C412yescomplemented
92#809—24no—
93#809—42no—
94#809—42no—
95C23×D4—31no—
96C22×D8—61no—
97C2×#121C2×D311yescharacteristic
98C2×C4×D4—21no—
99C22×C22⋊C4—61no—
100C22×C22⋊C4—61no—
101#447—34no—
102#447—62no—
103C2×#102—61no—
104C2×#102—61no—
105C2×#102—61no—
106C2×C22⋊C8—31no—
107C2×C22⋊C8—31no—
108#388—21no—
109#387—21no—
110#386—21no—
111#386—21no—
112#385—21no—
113#380C2×D311yescharacteristic
114C2×#95—31no—
115C2×#95C2×D311yescharacteristic
116#327—62no—
117C24×C3—21no—
118C24×C3C2×D411yescharacteristic
119C23×D3—22no—
120C23×D3—22no—
121C23×D3—22no—
122C23×D3—41no—
123C23×D3—41no—
124C23×D3—41no—
125C23×D3—41no—
126C23×D3—41no—
127C23×D3—41no—
128C23×D3—41no—
129C23×D3—41no—
130C23×D3—41no—
131C23×D3—41no—
132C23×D3—41no—
133C23×D3—41no—
134C2×C3×D4—22no—
135C2×C3×D4—22no—
136C2×C3×D4—21no—
137C2×C3×D4—41no—
138C2×C3×D4—41no—
139C2×C3×D4—41no—
140C2×C3×D4—41no—
141C2×C3×D4—41no—
142C2×C3×D4—41no—
143C2×C3×D4C22⋊C412yescomplemented
144C22×C3×C4—21no—
145C22×C3×C4—21no—
146C22×C3×C4—21no—
147C22×C3×C4—21no—
148C22×C3×C4—21no—
149C22×C3×C4—21no—
150C22×C3×C4—21no—
151C22×C3×C4—21no—
152C22×C3×C4—41no—
153C22×C3×C4—41no—
154C22×C3×C4—42no—
155C22×C3×C4—41no—
156C22×C3×C4—41no—
157C22×C3×C4—41no—
158C22×C3×C4—41no—
159C22×C3×C4—81no—
160C22×C3×C4—81no—
161C22×C3×C4D812yescomplemented
162C22×C3×C4C22×C411yescharacteristic
163C22×C3×C4C2×D411yescharacteristic
164C22×C3×C4C2×D411yescharacteristic
165C22×C3×C4C2×D411yescharacteristic
166C22×C3×C4#4111yescharacteristic
167C22×C3×C4#4111yescharacteristic
168C22×Dic3—22no—
169C22×Dic3—41no—
170C22×Dic3—41no—
171C22×Dic3—41no—
172C22×Dic3—41no—
173C22×Dic3—41no—
174C22×Dic3—41no—
175C2×D12—22no—
176C2×D12—22no—
177C2×D12—22no—
178C2×D12—22no—
179C2×D12—22no—
180C2×D12—22no—
181C2×D12—44no—
182C2×D12—44no—
183C2×D12—44no—
184C2×D12—41no—
185C2×D12—41no—
186C2×D12—41no—
187C2×D12—41no—
188C2×D12—44no—
189C2×D12—41no—
190C2×D12—41no—
191C3×C4⋊C4—21no—
192C3×C4⋊C4—41no—
193C3×C4⋊C4—41no—
194C3×C4⋊C4—42no—
195C3×C4⋊C4C22⋊C412yescomplemented
196C3×C22⋊C4—22no—
197C3×C22⋊C4—22no—
198C3×C22⋊C4—42no—
199C3×C22⋊C4—41no—
200C3×C22⋊C4—41no—
201C3×C22⋊C4—81no—
202C3×C22⋊C4—81no—
203C3×C42—42no—
204#213—48no—
205(C2×D3)⋊C4—48no—
206(C2×D3)⋊C4—48no—
207C2×C3⋊C8—22no—
208C2×C3⋊C8—41no—
209C2×C3⋊C8—41no—
210C2×C3⋊C8—41no—
211C2×C3⋊C8—41no—
212C2×C3⋊C8—41no—
213C25—61no—
214C22×D4—21no—
215C22×D4—32no—
216C22×D4—61no—
217C22×D4—61no—
218C22×D4—61no—
219C22×D4—61no—
220C22×D4—61no—
221C22×D4—122no—
222C22×D4—122no—
223C22×D4#6711yescharacteristic
224C23×C4—21no—
225C23×C4—61no—
226C23×C4#6711yescharacteristic
227C23×C4#6711yescharacteristic
228C23×C4C22×D311yescharacteristic
229C2×D8—64no—
230C2×D8—122no—
231C2×D8—122no—
232C22×C8—61no—
233C22×C8—61no—
234#121—21no—
235#121—21no—
236#121C4×D312yes—
237#121D1212yes—
238C4×D4—42no—
239C4×D4—42no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4#6711yescharacteristic
242C2×C4⋊C4#6711yescharacteristic
243C2×C4⋊C4#6711yescharacteristic
244C2×C22⋊C4—21no—
245C2×C22⋊C4—21no—
246C2×C22⋊C4—21no—
247C2×C22⋊C4—21no—
248C2×C22⋊C4—21no—
249C2×C22⋊C4—122no—
250C2×C22⋊C4—122no—
251C2×C22⋊C4—122no—
252C2×C22⋊C4—122no—
253C2×C22⋊C4—122no—
254C2×C22⋊C4—122no—
255C2×C42—21no—
256#102—64no—
257#102—122no—
258#102—122no—
259C22⋊C8—32no—
260C22⋊C8—61no—
261C22⋊C8—61no—
262C22⋊C8—61no—
263#95—21no—
264#95—21no—
265#95—21no—
266#95—21no—
267#95—21no—
268#95—61no—
269#95—61no—
270C23×C3—21no—
271C23×C3—21no—
272C23×C3—21no—
273C23×C3—21no—
274C23×C3—22no—
275C23×C3—41no—
276C23×C3—41no—
277C23×C3—41no—
278C23×C3—41no—
279C23×C3—41no—
280C23×C3—41no—
281C23×C3C4×D411yescharacteristic
282C23×C3#12011yescharacteristic
283C23×C3#12111yescharacteristic
284C23×C3C2×D811yescharacteristic
285C23×C3#13511yescharacteristic
286C23×C3AGL(1,ℤ/8)11yescharacteristic
287C23×C3AGL(1,ℤ/8)11yescharacteristic
288C22×D3—24no—
289C22×D3—42no—
290C22×D3—42no—
291C22×D3—42no—
292C22×D3—42no—
293C22×D3—42no—
294C22×D3—42no—
295C22×D3—42no—
296C22×D3—42no—
297C22×D3—42no—
298C22×D3—42no—
299C22×D3—42no—
300C22×D3—42no—
301C22×D3—42no—
302C22×D3—42no—
303C22×D3—42no—
304C22×D3—42no—
305C22×D3—42no—
306C22×D3—42no—
307C22×D3—81no—
308C22×D3—81no—
309C22×D3—81no—
310C22×D3—81no—
311C22×D3—81no—
312C22×D3—81no—
313C22×D3—81no—
314C22×D3—81no—
315C22×D3—81no—
316C22×D3—81no—
317C22×D3—81no—
318C22×D3—81no—
319C22×D3—81no—
320C22×D3—81no—
321C22×D3—81no—
322C22×D3—81no—
323C3×D4—42no—
324C3×D4—42no—
325C3×D4—42no—
326C3×D4—42no—
327C3×D4—42no—
328C3×D4—81no—
329C2×C3×C4—21no—
330C2×C3×C4—21no—
331C2×C3×C4—21no—
332C2×C3×C4—21no—
333C2×C3×C4—22no—
334C2×C3×C4—22no—
335C2×C3×C4—41no—
336C2×C3×C4—41no—
337C2×C3×C4—41no—
338C2×C3×C4—41no—
339C2×C3×C4—41no—
340C2×C3×C4—41no—
341C2×C3×C4—41no—
342C2×C3×C4—41no—
343C2×C3×C4—42no—
344C2×C3×C4—42no—
345C2×C3×C4—41no—
346C2×C3×C4—41no—
347C2×C3×C4—41no—
348C2×C3×C4—41no—
349C2×C3×C4—41no—
350C2×C3×C4—41no—
351C2×C3×C4—41no—
352C2×C3×C4—41no—
353C2×C3×C4—41no—
354C2×C3×C4—81no—
355C2×C3×C4—84no—
356C2×C3×C4—81no—
357C2×C3×C4C2×C22⋊C411yescharacteristic
358C2×C3×C4C4×D411yescharacteristic
359C2×C3×C4#12111yescharacteristic
360C2×C3×C4#12311yescharacteristic
361C2×Dic3—42no—
362C2×Dic3—42no—
363C2×Dic3—42no—
364C2×Dic3—81no—
365C2×Dic3—81no—
366C2×Dic3—81no—
367C2×Dic3—81no—
368D12—24no—
369D12—24no—
370D12—42no—
371D12—42no—
372D12—42no—
373D12—42no—
374D12—42no—
375D12—42no—
376C3⋊C8—42no—
377C3⋊C8—81no—
378C24—21no—
379C24—62no—
380C24—62no—
381C24—62no—
382C24—121no—
383C24—121no—
384C24—121no—
385C24—121no—
386C24—121no—
387C24—121no—
388C24—121no—
389C24—121no—
390C24—121no—
391C24—121no—
392C24—121no—
393C24—121no—
394C24C2×#6711yescharacteristic
395C2×D4—22no—
396C2×D4—22no—
397C2×D4—21no—
398C2×D4—41no—
399C2×D4—41no—
400C2×D4—41no—
401C2×D4—41no—
402C2×D4—41no—
403C2×D4—41no—
404C2×D4—62no—
405C2×D4—62no—
406C2×D4—62no—
407C2×D4—62no—
408C2×D4—62no—
409C2×D4—62no—
410C2×D4—121no—
411C2×D4—121no—
412C2×D4—121no—
413C2×D4—121no—
414C2×D4—124no—
415C2×D4—124no—
416C2×D4—124no—
417C2×D4—124no—
418C2×D4—121no—
419C2×D4—121no—
420C2×D4(C2×D3)⋊C412yescomplemented
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—41no—
430C22×C4—41no—
431C22×C4—42no—
432C22×C4—41no—
433C22×C4—41no—
434C22×C4—41no—
435C22×C4—41no—
436C22×C4—62no—
437C22×C4—81no—
438C22×C4—81no—
439C22×C4—121no—
440C22×C4—121no—
441C22×C4—121no—
442C22×C4—121no—
443C22×C4—121no—
444C22×C4—121no—
445C22×C4#21312yescomplemented
446C22×C4C2×C4×D311yescharacteristic
447C22×C4C2×D1211yescharacteristic
448C22×C4#23511yescharacteristic
449C22×C4#23511yescharacteristic
450C22×C4C2×#6711yescharacteristic
451C22×C4C2×#6711yescharacteristic
452D8—128no—
453C2×C8—62no—
454C2×C8—121no—
455C2×C8—121no—
456C2×C8—121no—
457C2×C8—121no—
458C2×C8—121no—
459C4⋊C4—21no—
460C4⋊C4—41no—
461C4⋊C4—41no—
462C4⋊C4—42no—
463C4⋊C4(C2×D3)⋊C412yescomplemented
464C22⋊C4—22no—
465C22⋊C4—22no—
466C22⋊C4—42no—
467C22⋊C4—41no—
468C22⋊C4—41no—
469C22⋊C4—81no—
470C22⋊C4—81no—
471C22⋊C4—128no—
472C22⋊C4—128no—
473C42—42no—
474C22×C3—21no—
475C22×C3—21no—
476C22×C3—21no—
477C22×C3—21no—
478C22×C3—21no—
479C22×C3—21no—
480C22×C3—21no—
481C22×C3—21no—
482C22×C3—21no—
483C22×C3—21no—
484C22×C3—21no—
485C22×C3—21no—
486C22×C3—42no—
487C22×C3—42no—
488C22×C3—42no—
489C22×C3—81no—
490C22×C3—81no—
491C22×C3—81no—
492C22×C3—81no—
493C22×C3#38611yescharacteristic
494C22×C3C4×D811yescharacteristic
495C22×C3#44211yescharacteristic
496C22×C3#44711yescharacteristic
497C22×C3#44911yescharacteristic
498C22×C3#45311yescharacteristic
499C22×C3#45411yescharacteristic
500C22×C3B(4,𝔽2)11yescharacteristic
501C22×C3B(4,𝔽2)11yescharacteristic
502C22×C3#45911yescharacteristic
503C22×C3#46311yescharacteristic
504C2×D3—44no—
505C2×D3—44no—
506C2×D3—44no—
507C2×D3—82no—
508C2×D3—82no—
509C2×D3—82no—
510C2×D3—82no—
511C2×D3—82no—
512C2×D3—82no—
513C2×D3—82no—
514C2×D3—82no—
515C2×D3—82no—
516C2×D3—82no—
517C2×D3—82no—
518C2×D3—82no—
519C3×C4—21no—
520C3×C4—21no—
521C3×C4—21no—
522C3×C4—21no—
523C3×C4—42no—
524C3×C4—42no—
525C3×C4—81no—
526C3×C4—81no—
527C3×C4—81no—
528Dic3—82no—
529C23—21no—
530C23—21no—
531C23—21no—
532C23—21no—
533C23—22no—
534C23—41no—
535C23—41no—
536C23—41no—
537C23—41no—
538C23—41no—
539C23—41no—
540C23—64no—
541C23—122no—
542C23—122no—
543C23—122no—
544C23—122no—
545C23—122no—
546C23—122no—
547C23—122no—
548C23—122no—
549C23—122no—
550C23—122no—
551C23—122no—
552C23—122no—
553C23—122no—
554C23—122no—
555C23—122no—
556C23—122no—
557C23—122no—
558C23—122no—
559C23—241no—
560C23—241no—
561C23—241no—
562C23—241no—
563C23—241no—
564C23—241no—
565C23—241no—
566C23—241no—
567C23—241no—
568C23—241no—
569C23—241no—
570C23—241no—
571C23—241no—
572C23—241no—
573C23—241no—
574C23—241no—
575C23C4×#6711yescharacteristic
576C23#93011yescharacteristic
577C23C2×#21311yescharacteristic
578C23#93211yescharacteristic
579C23#94911yescharacteristic
580C23#95011yescharacteristic
581C23#95311yescharacteristic
582D4—42no—
583D4—42no—
584D4—42no—
585D4—42no—
586D4—42no—
587D4—64no—
588D4—64no—
589D4—81no—
590D4—122no—
591D4—122no—
592D4—122no—
593D4—122no—
594D4—122no—
595D4—122no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—81no—
622C2×C4—84no—
623C2×C4—81no—
624C2×C4—122no—
625C2×C4—122no—
626C2×C4—122no—
627C2×C4—241no—
628C2×C4—241no—
629C2×C4—241no—
630C2×C4—241no—
631C2×C4C2×(C2×D3)⋊C411yescharacteristic
632C2×C4C4×#6711yescharacteristic
633C2×C4#92911yescharacteristic
634C2×C4#93011yescharacteristic
635C8—122no—
636C8—241no—
637C2×C3—21no—
638C2×C3—21no—
639C2×C3—21no—
640C2×C3—21no—
641C2×C3—82no—
642C2×C3#159511yescharacteristic
643C2×C3#159711yescharacteristic
644C2×C3#159811yescharacteristic
645C2×C3#160011yescharacteristic
646C2×C3#187911yescharacteristic
647C2×C3#188311yescharacteristic
648C2×C3#188911yescharacteristic
649D3—84no—
650C22—21no—
651C22—21no—
652C22—21no—
653C22—21no—
654C22—21no—
655C22—21no—
656C22—21no—
657C22—21no—
658C22—21no—
659C22—21no—
660C22—21no—
661C22—21no—
662C22—42no—
663C22—42no—
664C22—42no—
665C22—81no—
666C22—81no—
667C22—81no—
668C22—81no—
669C22—124no—
670C22—124no—
671C22—124no—
672C22—242no—
673C22—242no—
674C22—242no—
675C22—242no—
676C22—242no—
677C22—242no—
678C22—242no—
679C22—242no—
680C22—242no—
681C22—242no—
682C22—242no—
683C22—242no—
684C22C4×#21311yescharacteristic
685C22#501411yescharacteristic
686C22#501511yescharacteristic
687C22#501611yescharacteristic
688C22#521311yescharacteristic
689C22#521711yescharacteristic
690C22#523811yescharacteristic
691C22#524111yescharacteristic
692C22#524411yescharacteristic
693C22#524411yescharacteristic
694C22#524511yescharacteristic
695C4—21no—
696C4—21no—
697C4—21no—
698C4—21no—
699C4—42no—
700C4—42no—
701C4—81no—
702C4—81no—
703C4—81no—
704C4—242no—
705C3#904111yescharacteristic, complemented, Sylow
706C2—21no—
707C2—21no—
708C2—21no—
709C2—21no—
710C2—82no—
711C2—244no—
712C2#7161311yescharacteristic
713C2#7165411yescharacteristic
714C2#7166711yescharacteristic
715C2#7184511yescharacteristic
716C2#7187811yescharacteristic
717C2#7188411yescharacteristic
718C2#7188811yescharacteristic
subgroupG/H
centerC23#953
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#2325C2
socleC23×C3#120
prime radicalC2×#5036C2
cosocleC23×C3×C4C23
hypercenter#2325D3
hypocenterC3#9041
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2325

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C3×#2325C2semidirect
3. C2×#4537C2semidirect
4. #69521C2semidirect
5. #69426C2semidirect
6. #67624C2semidirect
7. #67613C2semidirect
26. #5036C4semidirect
32. #2325D3semidirect
53. C22×D12C2×C4semidirect
91. #830C2×C4semidirect
143. C2×C3×D4C22⋊C4semidirect
161. C22×C3×C4D8semidirect
195. C3×C4⋊C4C22⋊C4semidirect
420. C2×D4(C2×D3)⋊C4semidirect
445. C22×C4#213semidirect
463. C4⋊C4(C2×D3)⋊C4semidirect
705. C3#9041semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C3×#23252
C2×#50362
C2×#45372
#695212
#694262
#676242
#676132
#90413